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In many systems consisting of interacting subsystems, the complex
interactions between elements can be represented using multilayer
networks. However percolation, key to understanding connectivity
and robustness, is not trivially generalised to multiple layers.
This Element describes a generalisation of percolation to
multilayer networks: weak multiplex percolation. A node belongs to
a connected component if at least one of its neighbours in each
layer is in this component. The authors fully describe the critical
phenomena of this process. In two layers with finite second moments
of the degree distributions the authors observe an unusual
continuous transition with quadratic growth above the threshold.
When the second moments diverge, the singularity is determined by
the asymptotics of the degree distributions, creating a rich set of
critical behaviours. In three or more layers the authors find a
discontinuous hybrid transition which persists even in highly
heterogeneous degree distributions, becoming continuous only when
the powerlaw exponent reaches $1+1/(M-1)$ for $M$ layers.
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